Intra species factors calculation

There is variation between individuals concerning their individual sensitivities to experience health effects. In some scenarios the aim is to perform assessments for the sensitive individuals instead of the average individuals for which the points of departure are derived. If this is the case, then extrapolation is required to translate hazard characterisations derived for the average individual to hazard characterisations for a sensitive individual.

Traditionally a fixed safety factor describes the variation between individuals. Little information is available about the true distribution of human sensitivities, so there is also a large uncertainty. In MCRA, the intra-species variability is modelled explicitly using a lognormal distribution, characterised by a geometric mean (GM) equal to 1 and a geometric standard deviation (GSD). This distribution is used to sample individual hazard characterisations. This effectively converts the description of hazard characterisations to include variability, with an unbiased central value. GM is 1 by definition (50% of the population is assumed to be less sensitive and 50% more sensitive than the average individual) and has no uncertainty. On the other hand, there is uncertainty about the GSD or SD = log(GSD). This uncertainty is described by a chi-square distribution with $\mathit{df}$ degrees of freedom:

$$\frac{df\cdot S{D}^2}{{\sigma }^2}\sim {\chi }_{df}^2$$

where $\sigma$ is the true standard deviation and ${\chi }_{df}^2$ denotes a chi-square distribution with $\mathit{df}$ degrees of freedom. Thus $\mathit{df}$ characterises the amount of uncertainty regarding the intra species variation quantified by (G)SD.

It is difficult to specify values for GSD and $\mathit{df}$. A practical way to do this is, define a p95-sensitive individual having an intra-species factor corresponding to the p95 percentile of the distribution describing the variability.

Assume that the p95-sensitive individuals are between e.g. 2 and 10 times more sensitive than the average human. The values 2 and 10 are to be interpreted as uncertainty bounds, the 2.5th and 97.5th percentiles of the uncertainty distribution for the $F_{intra,p95}$.

Then, p95-sensitive individuals are between $F_{intra,p95p2.5}$ and $F_{intra,p95p97.5}$ more sensitive than the average human, where the first index indicate the variability distribution and the second the uncertainty distribution.

For the lognormal distribution we have:

$$ln(F_{intra,p95})=ln(GM)+1.645\cdot \sigma =1.645\cdot \sigma$$

And thus $\sigma =\frac{ln(F_{intra,p95})}{1.645}$. The uncertainty statement can be re-expressed as a 95% confidence interval for $\sigma$:

$$\frac{ln(F_{intra,p95p2.5})}{1.645}\le \sigma \le \frac{ln(F_{intra,p95p97.5})}{1.645}$$

A standard two-sided 95% confidence interval for $\sigma$ derived from the ${\chi }^2$ distribution is:

$$SD\cdot \sqrt{\frac{df}{{\chi }_{df,0.975}^2}}\le \sigma \le SD\cdot \sqrt{\frac{df}{{\chi }_{df,0.025}^2}}$$

Equating the two lower bounds as well as the two upper bounds we obtain two equations with two unknowns. Substituting for SD and rearranging we find that $df$ can be found from:

$$\frac{{\chi }_{df,0.975}^2}{{\chi }_{df,0.025}^2}={\left(\frac{ln(F_{intra,p95p97.5})}{ln(F_{intra,p95p2.5})}\right)}^2$$

And then GSD can be calculated as:

$$GSD=exp(SD)=(F_{intra,p95p97.5})\sqrt{\frac{{\chi }_{df,0.025}^2}{df}}/1.645$$

A bisection algorithm can be used to find the value for $df$ for which the chi-square ratio in the example is

$$\frac{{\chi }_{df,0.975}^2}{{\chi }_{df,0.025}^2}={\left(\frac{ln(10)}{ln(2)}\right)}^2=11.035$$

This value is attained for $df$ = 6.25. See also van der Voet et al. (2009)